The Lagrange interpolation problem on the real line has been widely studied for a long time and many results about convergence are known, (see [1]}, [2]}, [3]} and [4]}). If we only assume the continuity of the function, it is well known that the behavior is rather irregular. Faber has proved that for each nodal system there exists a continuous function such that the sequence of Lagrange interpolation polynomials is not uniformly convergent. Bernstein has also proved the existence of a continuous function such that the sequence of Lagrange interpolation polynomials is unbounded on a prefixed point. In the case of the nodal systems constituted by the zeros of the Tchebychef polynomials of the first kind, many results are known. Although these last nodal systems are good for interpolation, Grünwald in [5]} and Marcinkiewicz in [6]} have proved the existence of a continuous function such that the sequence of Lagrange interpolation polynomials, corresponding to the Tchebychef nodal system, is divergent. After this result a natural problem was to obtain an analogous result for an arbitrary nodal system. This result was obtained by Erdös and Vértesi in [7]}, where they prove that for each nodal system on \([-1,1]\)  there exists a continuous function such that the sequence of Lagrange interpolation polynomials diverges for almost every point in \([-1,1]\) .
Thus, to obtain better properties about the convergence of the sequence of Lagrange interpolation polynomials, it is needed to impose some restriction to the function, such as, a condition on its modulus of continuity. In the case of Jacobi abscissas, Szegő has obtained important results about convergence by imposing some conditions to the modulus of continuity of the function, (see [8]}). For example, in the case of the Tchebychef abscissas of first kind, he obtained the uniform convergence to the function on \([-1,1]\) , under the assumption that its modulus of continuity is \(o(\vert \log \delta \vert ^{-1})\) . Szegő has also obtained uniform convergence of the sequence of Lagrange interpolation polynomials for more general nodal systems, under the assumptions that the nodes are the zeros of the orthogonal polynomials with respect to a weight function \(w(x)\)  such that \(w(x)\sqrt{1-x^2} \ge \mu >0 ,\; x \in (-1,1)\)  and the modulus of continuity of the functions is \(o(\delta ^{\frac{1}{2}})\)  with
\( \delta \rightarrow 0.\)
